Решебник по алгебре 8 класс Мерзляк ФГОС Задание 732

 

\[\boxed{\mathbf{732\ (732).\ }Еуроки\ - \ ДЗ\ без\ мороки}\]

\[x^{2} + 5x - 16 = 0\]

\[x_{1} + x_{2} = - 5;\ \ \ \ x_{1} \cdot x_{2} = - 16\]

\[1)\text{\ x}_{1}²x_{2} + x_{1}x_{2}² =\]

\[= x_{1}\left( x_{1}x_{2} + x_{2}^{2} \right) =\]

\[= x_{1}x_{2}\left( x_{1}{+ x}_{2} \right) = - 16 \cdot ( - 5) =\]

\[= 80\]

\[2)\ \frac{x_{2}}{x_{1}} + \frac{x_{1}}{x_{2}} = \frac{x_{2}^{2} + x_{1}^{2}}{x_{1}x_{2}} =\]

\[= \frac{x_{2}^{2} + x_{1}^{2} + 2x_{1}x_{2} - 2x_{1}x_{2}}{x_{1}x_{2}} =\]

\[= \frac{\left( x_{1} + x_{2} \right)^{2} - 2x_{1}x_{2}}{x_{1}x_{2}} =\]

\[= \frac{25 + 32}{- 16} = - \frac{57}{16} = - 3\frac{9}{16}\]

\[3)\ \left| x_{2} - x_{1} \right| = \sqrt{\left( x_{2} - x_{1} \right)^{2}} =\]

\[= \sqrt{\left( x_{2}^{2} + x_{1}^{2} - 2x_{1}x_{2} \right)} =\]

\[= \sqrt{\left( x_{1} + x_{2} \right)^{2} - 4x_{1}x_{2}} =\]

\[= \sqrt{25 - 4 \cdot ( - 16)} = \sqrt{89}\ \]

\[\boxed{\mathbf{73}\mathbf{2}\mathbf{\text{.\ }}Еуроки\ - \ ДЗ\ без\ мороки}\]

\[Уравнение\ имеет\ один\ корень\ \]

\[при\ D < 0.\]

\[1)\ bx² - 6x - 7 = 0\]

\[D = 36 + 28b = 0\]

\[28b = - 36\]

\[b = - \frac{36}{28} = - \frac{9}{7} = - 1\frac{2}{7}\]

\[b = 0:\]

\[\ - 6x - 7 = 0 \Longrightarrow \ \ x = - \frac{7}{6}\]

\[Ответ:при\ b = - 1\frac{2}{7};\ \ \ b = 0.\]

\[2)\ (b + 5)x² - (b + 6)x + 3 = 0\]

\[D = (b + 6)^{2} - 12 \cdot (b + 5) =\]

\[= b^{2} + 12b + 36 - 12b - 60 =\]

\[= b^{2} - 24 = 0\]

\[b^{2} = 24\]

\[b = 2\sqrt{6}\]

\[b = - 2\sqrt{6}\]

\[b + 5 = 0 \Longrightarrow \ \ b = - 5\]

\[Ответ:при\ b = 2\sqrt{6};\ \ \]

\[b = - 2\sqrt{6};\ - 5.\]

\[3)\ (b - 4)x² + (2b - 8)x + 15 =\]

\[= 0\]

\[D = (2b - 8)^{2} - 60 \cdot (b - 4) =\]

\[= 4b^{2} - 92b + 304 = 0\]

\[4b^{2} - 92b + 304 = 0\]

\[b^{2} - 23b + 76 = 0\]

\[D = 529 - 4 \cdot 76 = 225\]

\[b = \frac{23 \pm 15}{2}\]

\[b = 19\]

\[b = 4\ \]

\[2b - 8 \neq 0 \Longrightarrow \ \ b \neq 4\]

\[Ответ:при\ b = 19.\]